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%I #3 Mar 30 2012 17:36:13
%S 1,0,1,0,1,1,0,1,2,1,0,2,2,3,1,0,4,4,4,4,1,0,8,8,8,7,5,1,0,17,16,17,
%T 14,11,6,1,0,37,34,36,31,23,16,7,1,0,82,74,79,68,53,36,22,8,1,0,185,
%U 164,177,152,121,86,54,29,9,1,0,423,370,402,346,278,204,134,78,37,10,1,0
%N Triangle read by rows: T(n,k) is the number of peakless Motzkin paths of length n and having k steps that touch the x-axis (1<=k<=n).
%C T(n,k)=number of secondary structures of size n in which the shortest path from one end to the other one has length k-1. Row sums yield A004148. T(n,2)=A004148(n-2). T(n,3)=2*A004148(n-3) for n>=4. Sum(k*T(n,k),k=1..n)=A128096(n).
%F G.f.=2/[2-2tz-t^2+t^2*z+t^2*z^2+t^2*sqrt((1+z+z^2)(1-3z+z^2))]-1.
%e T(5,4)=3 because we have HU(H)DH, HHU(H)D and U(H)DHH, where U=(1,1), H=(1,0) and D=(1,-1) and the steps that do not touch the x-axis are shown between parentheses.
%e Triangle starts:
%e 1;
%e 0,1;
%e 0,1,1;
%e 0,1,2,1;
%e 0,2,2,3,1;
%e 0,4,4,4,4,1;
%e 0,8,8,8,7,5,1;
%p G:=2/(2-2*t*z-t^2+t^2*z+t^2*z^2+t^2*sqrt((1+z+z^2)*(1-3*z+z^2)))-1: Gser:=simplify(series(G,z=0,15)): for n from 1 to 13 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 1 to 13 do seq(coeff(P[n],t,j),j=1..n) od; # yields sequence in triangular form
%Y Cf. A004148, A128096.
%K nonn,tabl
%O 1,9
%A _Emeric Deutsch_, Feb 14 2007
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